Calculus Examples

$\ln (\cos^{2} (x))$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \cos^{2} (x)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\cos^{2} (x)$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\cos^{2} (x)]$

Step 1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{1}{u_{1}} \frac{d}{d x} [\cos^{2} (x)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\cos^{2} (x)$ .

$\frac{1}{\cos^{2} (x)} \frac{d}{d x} [\cos^{2} (x)]$

Step 2

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

$\sec^{2} (x) \frac{d}{d x} [\cos^{2} (x)]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cos (x)$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $\cos (x)$ .

$\sec^{2} (x) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\cos (x)])$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$\sec^{2} (x) (2 u_{2} \frac{d}{d x} [\cos (x)])$

Step 3.3

Replace all occurrences of $u_{2}$ with $\cos (x)$ .

$\sec^{2} (x) (2 \cos (x) \frac{d}{d x} [\cos (x)])$

Step 4

Move $2$ to the left of $\sec^{2} (x)$ .

$2 \cdot \sec^{2} (x) \cos (x) \frac{d}{d x} [\cos (x)]$

Step 5

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$2 \sec^{2} (x) \cos (x) (- \sin (x))$

Step 6

Multiply $- 1$ by $2$ .

$- 2 \sec^{2} (x) \cos (x) \sin (x)$

Step 7

Simplify.

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Step 7.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$- 2 {(\frac{1}{\cos (x)})}^{2} \cos (x) \sin (x)$

Step 7.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

$- 2 \frac{1^{2}}{\cos^{2} (x)} \cos (x) \sin (x)$

Step 7.3

One to any power is one.

$- 2 \frac{1}{\cos^{2} (x)} \cos (x) \sin (x)$

Step 7.4

Combine $- 2$ and $\frac{1}{\cos^{2} (x)}$ .

$\frac{- 2}{\cos^{2} (x)} \cos (x) \sin (x)$

Step 7.5

Cancel the common factor of $\cos (x)$ .

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Step 7.5.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\frac{- 2}{\cos (x) \cos (x)} \cos (x) \sin (x)$

Step 7.5.2

Cancel the common factor.

$\frac{- 2}{\cos (x) \cos (x)} \cos (x) \sin (x)$

Step 7.5.3

Rewrite the expression.

$\frac{- 2}{\cos (x)} \sin (x)$

Step 7.6

Move the negative in front of the fraction.

$- \frac{2}{\cos (x)} \sin (x)$

Step 7.7

Combine $\sin (x)$ and $\frac{2}{\cos (x)}$ .

$- \frac{\sin (x) \cdot 2}{\cos (x)}$

Step 7.8

Move $2$ to the left of $\sin (x)$ .

$- \frac{2 \cdot \sin (x)}{\cos (x)}$

Step 7.9

Separate fractions.

$- (\frac{2}{1} \cdot \frac{\sin (x)}{\cos (x)})$

Step 7.10

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$- (\frac{2}{1} \tan (x))$

Step 7.11

Divide $2$ by $1$ .

$- (2 \tan (x))$

Step 7.12

Multiply $2$ by $- 1$ .

$- 2 \tan (x)$